Integral Menger Curvature

Transcription

1 University of Washington December 1, 2016

2 Menger Curvature Given 3 distinct points x, y, z C, one can define the Menger Curvature, denoted c(x, y, z) by c(x, y, z) = 1 R(x, y, z), where R(x, y, z) is the radius of the unique circle containing x, y, z.

3 In the case that the three points are colinear, we say that R(x, y, z) = + so c(x, y, z) = 0. On the other hand...

4 Inscribed angle Fact: Given a circle, and any two distinct points on it, any inscribed angle is half the central angle. Figure: Picture courtesy of Wikipedia

5 The Law of Sines x_1 a_1 x_2 x_0 x_0 - x_2 Theorem If x 0, x 1, x 2 are the verticies of a triangle, and α i is the angle at vertex x i then x 1 x 2 sin(α 0 ) = x 2 x 0 sin(α 1 ) = x 1 x 0 sin(α 2 ) = d where d is the diameter of the circumcircle.

6 Menger Curvature Revisited Fact: If T is the area of a triangle with vertices x 0, x 1, x 2 C then c(x, y, z) = 4T x 2 x 0 x 2 x 1 x 1 x 0

7 Hausdorff n-measure For those unfamiliar with the Hausdorff n-measure, it turns out to have many nice properties. Throughout this talk, we will consider a Borel measure H n as a set function on R N for n {0, 1,..., N}. H N is a multiple of the Lebesgue measure on R N. H N 1 coincides with (a multiple of) surface area in the usual sense. H 1 coincides with (a multiple of) length in the usual sense. In fact, informally speaking H n coincides with (a multiple of) what you would hope the n-dimensional volume of an object in R N would be.

8 Despite the usage in this talk, one can consider the Hausdorff s-measure when s is any real number. If E R N with N N then H s (E) = lim δ 0 H s δ(e), where Hδ(E) s = inf diam(a j ) s E j=1 j A j and diam(a j ) < δ. H s is called the s-dimensional Hausdorff measure on R N. It turns out that for a set E R N the number α defined by α = sup{s : H s (E) = 0} = inf{s : H s (E) = + } exists, 0 α N and is called the Hausdorff dimension of E.

9 Regularity of n-dimensional sets A Borel set E R N is said to be n-rectifiable if there exists a collection of Lipschitz functions {f i } i N such that f i : R n R N satisfy H n ( E \ ( i N f i (R n ) )) = 0. Sean gave us many examples of why rectifiable sets are interesting and important. A set E R N is said to be n-ahlfors regular if E is Borel and for every x E and every r > 0 small enough: cr n H n (E B(x, r)) Cr n.

10 Total Menger curvature and 1-rectifiability We define the total Menger curvature of E R N by M c 2(E) = c 2 (x, y, z) dh 1 (x) dh 1 (y) dh 1 (z) E E E Theorem (Leger 99) If E R N is a 1-Ahlfors regular set and M c 2(E) <, then E is 1-rectifiable. In fact, in the same work, Leger proved that if E R N is (N 1)-rectifiable, and M c 2(E) < then E is (N 1)-rectifiable. It may be worth noting that David proved something similar in C first. (As far as I can tell it was not published.) It was known that David s proof did not generalize to higher dimensions.

11 Meurer s Main result Theorem (Meuerer 2015) Let E R N be a borel set with M K 2(E) <, where K 2 is any proper integrand. Then E is n-rectifiable. K 2 -proper integrand requires that E is n-upper-ahlfors regular. It does NOT require lower-ahlfors regularity. Theorem (Leger 99) Let n {1, N 1} and borel E R N be a n-ahlfors regular set satisfying M c 2(E) <. Then E is n-rectifiable.

12 Simplices: definitions, notation, and notions. Given n + 1 points (x 0,..., x n ) in R N we can define the simplex (x 0,..., x n ) to be the convex-hull of the points {x 0,..., x n }. This simplex is called an n-simplex if H n ( (x 0,..., x n )) > 0. For n = 2, an n-simplex is a triangle, for n = 3 a tetrahedron.

13 Define fc i ( (x 0,..., x n )) = (x 0,..., x i 1, x i+1,..., x n ) Define h i ( (x 0,..., x n )) = dist(x i, aff({x 0,..., x i 1, x i+1,..., x n }) An n-simplex is called an (n, C)-simplex if h i ( (x 0,..., x n )) C for all i = 0,..., n.

14 Example of K 2 -proper integrand Define K(x 0,..., x n+1 ) = Hn+1 ( (x 0,..., x n+1 )) 0 i<j n+1 d(x i, x j ) The numerator denotes the (n + 1)-dimensional volume of the (n + 1)-simplex. When n = 1, the numerator is the area of a triangle, so... H 2 ( (x 0, x 1, x 2 )) K(x 0, x 1, x 2 ) = x 0 x 1 x 0 x 2 x 1 x 2 = c(x 0, x 1, x 2 ). 4 In particular, with this integrand, and restricting our attention to n = 1, Meurer s theorem is the same as Leger s theorem EXCEPT that it does not require the lower-ahlfors regularity of the set E.

15 K p -proper integrand Given µ an upper-ahlfors regular, borel measure with compact support a function K : (R N ) n+2 [0, ), the function K p is called a µ-proper integrand if: K is a (µ) n+2 -measurable function. There exists constants α, l 1 such that for all t > 0, C 1, x R N and all (n, t C )-simplices (x 0,..., x n ) B(x, Ct) we have: ( dist(w, aff(x0,..., x n ) for all w B(x, Ct) t ) p ( αc l K p x0 t,..., x n t, w ) t For all t > 0 we have K p ( x0 t,..., xn+1 t ) = t n(n+1) K p (x 0,..., x n+1 ). For every b R N, we have K(x 0 + b,..., x n+1 + b) = K(x 0,..., x n+1 ).

16 Then, thinking of K as being a higher-dimensional analog of Menger curvature c. Theorem (Meurer 2015) If E R N is borel, E is n-upper ahlfors regular for n {1,..., N 1}, and M K 2(E) = K 2 d(h n ) n+2 <, E n+2 then E is n-rectifiable. Theorem (Leger 1999) If E R N is borel, E is 1-ahlfors regular and M c 2(E) = c 2 dh 1 dh 1 dh 1 <, then E is 1-rectifiable. E E E

17 Big step in Meurer s paper Theorem (Meurer 2015) If K p is a symmetric µ-proper integrand, and 0 < λ < 2 n, k > 2, k 0 1. There exist constants k 1 > 5 and C 1 such that if x R N satisfies µ(b(x, t)) t n then for every y B(x, k 0 t) we have > λ β p;k (y, t) p C M K p ;k 1 (x, t) t n C MKp ;k 1+k 0 (y, t) t n That is, Jones β-numbers are pointwise bounded by the local Menger curvature.

18 Local Menger Curvature and β-numbers Recall the β-numbers β p;k (x, t) p = inf P P(N,n) 1 t n B(x,kt) ( ) p dist(y, P) dµ(y) Define Local Menger Curvature M Kp ;k(x, t) = K p (x 0,..., x n+1 ) dµ(x 0 )... dµ(x n+1 ) (O k (x,t)) n+2 where O k (x, t) = {(x 0,..., x n+1 ) (B(x, kt)) n+2 d(xi, x j ) tk } i j t

19 How local Menger curvature bounds β-numbers. The lower-density condition: µ(b(x, t)) t n > λ is used to find an (n + 1)-simplex T = (x 0,..., x n+1 ) spt(µ) B(x, t) such that each fc i (T ) is an (n, tα)-simplex for α-small enough, with an additional lower bound on the measure of a small ball near the corner. In essence, the lower-density condition is just used to find at least (n + 2) points in B(x, t) that are well-spaced and each one is near a large piece of the measure µ B(x, t).

20 From here, Meurer proves the existence of points {z i } n+1 i=0 where z i is near the corner x i such that: 1 (z0,...,ẑ l,...,z n+1,w) O k (x,t)k p (z 0,..., ẑ l,..., z n+1, w) dµ(w) C M K p ;k 1 (x, t) t (n+1)n, which basically says that for any (n + 1) points chosen from {z j } n+1 j=0, the cross-sectional local Menger curvature is smaller than the local Menger curvature, up to a constant and a scaling by t.

21 If for j = 0,..., n + 1 we define P j = aff{z 0,..., ẑ j,..., z n+1 }, then ( ) p dist(zn+1, P n+1 ) C M K p ;k 1 (x, t) t despite the fact that fc i (T ) is an (n, tα)-simplex for all i {0,..., n + 1}. Remember the goal β p;k (y, t) p C M K p ;k 1 (x, t) t n C MKp ;k 1+k 0 (y, t) t n t n

22 Because of the technical and annoying second part of the definition of a µ-proper integrand, it follows that for any w B(x, (k + k 0 )t) ( ) p ( dist(w, aff(z0,..., ẑ j,..., z n+1 )) K p z0 t t,..., ẑj t,..., z n+1, w ) t t

23 Then, by integrating the previous estimate on the swiss-cheese space O k (x, t) we discover that for all w B(x, (k + k 0 )t) that are sufficiently far from {x i } B(x,(k+k 0)t)\ n+1 j=0 2B j ( dist(w, Pn+1 ) t ) p dµ(w) M K p;k1 (x, t) We also get 2B j ( dist(w, Pj ) t ) p dµ(w) M K p;k1 (x, t).

24 To remedy this situation, we can consider w = π Pj (w). Then, ( ) [ p (dist(w, ) p ( dist(w, Pn+1 ) 2 p 1 Pj ) dist(w ) ] p, P n+1 ) +, t t t where the latter term can actually be bounded by a constant multiple of ( ) p dist(zn+1, P n+1 ) M K p;k1 (x, t), t which is why we worked so hard to choose z n+1 to satisfy the inequality.

25 All in all, we found that for the particular plane P n+1 ( ) p 1 dist(w, Pn+1 ) t n dµ(w) M K p ;k 1 (x, t) B(x,(k+k 0)t) t t n But, k + k 0 > k guarantees that β p;k (x, t) p β p;k+k0 (x, t) p 1 ( ) p dist(w, Pn+1 ) t n dµ(w) B(x,(k+k 0)t) t So we win: β p;k (x, t) p M K p ;k 1 (x, t) t n

26 Where to go from here? It s known that when you control δ 0 β2 (x, t) dt t decomposition is possible. that a Corona-like Meurer only has a pointwise bound on β-numbers and not a bound on this integral. But, nonetheless, a Corona-like decomposition still exists. In fact, Meurer goes on to prove the existence of a Corona decomposition when p = 2 with the result that you can cover 99% of spt(µ) B(x, t) with a single Lipschitz graph.

27 Questions from here: Covering 99% of spt(µ) B(x, t) is a lot. Can you in fact get uniform rectifiability? (Apparently yes, if we add lower Ahlfors regularity to our hypothesis.) Since K(x 0,..., x n+1 ) = Hn (x 0,...,x n+1) 0 i<j n x j x i is supposed to be a curvature, it should tell us about the geometry of a set up to second order. Rectifiability is only up to first order. So, what more could we learn by finite (for instance, would M K 2(E) < actually imply E is bi-lipschitz parametrizeable?) In the spirit of Leger s result: would Meurer s proof also work if n were replaced with N n in the definition of proper integrand? Can one find a formula for the radius of the circumsphere in terms of distances between corners that relates the generalized Menger Curvature to the L 2 -boundedness of the Riesz transform? (Similar to what Melnikov did with the classical Menger Curvature and the Cauchy transform.)